### Abstract

Let R be a prime ring and set [x, y]_{1} = [x, y] = xy - yx for x, y ∈ R and inductively [x, y]_{k} = [[x, y]_{k-1}, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x]_{k} = 0 for all x ∈ L, where k is a fixed positive integer, then charR = 2 and for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.

Original language | English |
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Pages (from-to) | 259-270 |

Number of pages | 12 |

Journal | Monatshefte fur Mathematik |

Volume | 158 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Monatshefte fur Mathematik*,

*158*(3), 259-270. https://doi.org/10.1007/s00605-008-0043-5

}

*Monatshefte fur Mathematik*, vol. 158, no. 3, pp. 259-270. https://doi.org/10.1007/s00605-008-0043-5

**An engel condition with skew derivations.** / Chou, Ming Chu; Liu, Cheng Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An engel condition with skew derivations

AU - Chou, Ming Chu

AU - Liu, Cheng Kai

PY - 2009/1/1

Y1 - 2009/1/1

N2 - Let R be a prime ring and set [x, y]1 = [x, y] = xy - yx for x, y ∈ R and inductively [x, y]k = [[x, y]k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x]k = 0 for all x ∈ L, where k is a fixed positive integer, then charR = 2 and for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.

AB - Let R be a prime ring and set [x, y]1 = [x, y] = xy - yx for x, y ∈ R and inductively [x, y]k = [[x, y]k-1, y] for k > 1. We apply the theory of generalized polynomial identities with automorphisms and skew derivations to obtain the following result: If δ is a nonzero σ-derivation of R and L is a noncommutative Lie ideal of R so that [δ(x), x]k = 0 for all x ∈ L, where k is a fixed positive integer, then charR = 2 and for some field F. This result generalizes the case of derivations by Lanski and also the case of automorphisms by Mayne.

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UR - http://www.scopus.com/inward/citedby.url?scp=70350367994&partnerID=8YFLogxK

U2 - 10.1007/s00605-008-0043-5

DO - 10.1007/s00605-008-0043-5

M3 - Article

AN - SCOPUS:70350367994

VL - 158

SP - 259

EP - 270

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 3

ER -